All of the 5th grade teachers and students from Springer went on a field trip to an art museum. Tickets were $$5.00$ each for teachers and $$3.50$ each for students, and the group paid $$43.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$15.00$ each for teachers and $$8.50$ each for students, and the group paid $$113.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+3.5y = 43}$ ${15x+8.5y = 113}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-15x-10.5y = -129}$ ${15x+8.5y = 113}$ Add the top and bottom equations together. $ -2y = -16 $ $ y = \dfrac{-16}{-2}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {5x+3.5y = 43}$ to find $x$ ${5x + 3.5}{(8)}{= 43}$ $5x+28 = 43$ $5x = 15$ $x = \dfrac{15}{5}$ ${x = 3}$ You can also plug ${y = 8}$ into $ {15x+8.5y = 113}$ and get the same answer for $x$ ${15x + 8.5}{(8)}{= 113}$ ${x = 3}$ There were $3$ teachers and $8$ students on the field trips.